Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×C5×C4⋊C4 ⋊ C2 |
| Sylow | C5 ⋊ #2651 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D5 |
| commutator, socle, cosocle | C23×C5 | C24 |
| Fitting | C4×C5×C4⋊C4 | C2 |
| prime radical | C22×D20 | C22 |
| hypercenter | C4×C4⋊C4 | D5 |
| hypocenter | C5 | #2651 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
420 conjugacy classes of subgroups.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^7,5|a^2=e,b^2=eg,d^2=f,b^a=beg,c^a=cf,c^b=cg,d^c=deg,h^c=h^4112 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#2651, C2×#10605546, #12266000, … |
| maximal normal subgroup of | D5×#2651, C2×#10605546, #12266000, … |
| maximal quotient of | #12025769, #12025858, #12025918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 7628);